Paper container and method of manufacturing it

ABSTRACT

It is an object of the present invention to provide a method of calculating a development plan of a paper container of deep bottom integrally formed from a single-sheet blank.  
     According to the present invention, in order to achieve the above object, an annular rule line  6  constituting a regular polygonal shape is formed at the center of a single-sheet blank to constitute the bottom face of the paper container, and divided faces  5  to constitute the outside of the peripheral face of the paper container are formed on the outside of the annular rule line  6 . The blank portions between the divided faces  5  constitute inner pleated faces  4 . Each of the blank portions is folded downwards along the rule line  7  and folded upwards along the line  9 , so that the blank portion is folded to define two triangles  8  with an angle φ and the overlapping portions thus obtained constitute an inner wall face  4 . The lateral edges of the divided faces  5  are brought together by folding up the annular rule line  6  while folding the inner pleated faces  4  in two along the lines of symmetry  7  and  9 , and the inner pleated faces are overlapped onto the divided faces, whereby a paper container is manufactured.

BACKBROUND OF THE INVENTION

[0001] 1. Field of the Invention

[0002] The present invention relates to a paper container and method ofmanufacturing it that is used as a container for food products or plantpot etc. In more detail, it relates to a paper container and method ofmanufacturing it having a deep bottom and formed by folding asingle-sheet blank.

[0003] 2. Description of the Related Art

[0004] Conventionally, for the distribution of food products etc,plastic containers, which are easily molded, are frequently used.However, recently, on account of problems concerning elution ofenvironmental hormones or disposal processing after use, the use ofpaper containers is being re-evaluated. As methods of manufacturingpaper containers, the method of sticking together and the papermakingmethod etc are well known. In the former i.e. the sticking-togethermethod, for example raw-material paper sheets that have been subjectedto laminating processing are employed to separately mould blanks whichare used for forming the trunk and the bottom of the container; thesetwo are then united by hot pressure fixing in a metal mold.

[0005] In the latter i.e. the paper-making method, the paper fibers aredispersed in water and the basic shape of the container is produced byfiltering this colloidal solution using a paper-making mesh ofprescribed shape and dewatering; the paper container is thenmanufactured by hot pressing or by drying this using a current of hotair. These methods had the drawbacks that the number of steps necessarywas large, making them costly, and that the containers obtained hadlittle resistance to water and so could not be employed for containersthat need to be waterproof, such as containers for drinks or plant pots.

[0006] Also, the drawing method of integrally forming a paper containerfrom a single-sheet blank is conventionally known and is commonlyemployed. With this drawing method, waterproof containers can bemanufactured efficiently and at low cost by for example using blanksthat have been subjected to laminating processing.

[0007] This drawing method has the advantage that a waterproof productcan be produced comparatively easily with a small number of steps, sinceit is integrally formed from a single-sheet blank. However, setting theconditions for the processing is extraordinarily difficult and inparticular there was the difficulty that the blank tended to tear in thecase of deep drawing. Consequently, conventional paper containersobtained by drawing were of shallow bottom, which restricted theirapplication.

[0008] The present invention was made in view of the technicalbackground described above and achieves the following object.

SUMMARY OF THE INVENTION

[0009] An object of the present invention is to provide a papercontainer of deep bottom integrally formed from a single-sheet blank,and a method of manufacturing it.

[0010] In a method of manufacturing a paper container of deep bottomintegrally formed from a single-sheet blank, a further object of thepresent invention is to provide a method of calculating the developmentplan of the paper container.

[0011] In order to achieve the above object of the present invention, amethod of manufacturing a paper container is provided wherein a blank isobtained by cutting a single-sheet of raw-material paper to a prescribedshape and an annular rule line constituting a regular polygonal shape isformed in the middle of this blank and designated as the bottom face ofthe paper container. After this, divided faces on the outside of theperipheral wall face constituting the peripheral wall face of the papercontainer and inner pleated faces on the inside are formed on theoutside of the annular rule line. The divided faces are of the samenumber as the number of corners of the bottom face, and are arranged toextend from each side of the annular rule line to the outside. The blankregions between the divided faces constitute the inner pleated faces,the inner wall faces being bisected by axes of symmetry extendingdividing the inner pleated faces into two symmetrical portions from thecorners of the annular rule line. After this, the inside edges of eachdivided face are brought together by folding the annular rule line whilefolding each inner pleated face in two along the axis of symmetry, andthe region inside the annular rule line is made to constitute the bottomface by folding over the inner pleated faces on each divided face.

[0012] If the height of the paper container, the radius of the uppermostface of the paper container, the radius of the lowermost face of thepaper container, and the number of corners of the bottom face of thepaper container are determined, a paper container of any desired shapewith an open upper surface can be produced. The condition of the paperat the rim of the uppermost face of the paper container can be made tobe a single sheet, or three sheets, or, if appropriate, five sheets, atparticular locations.

[0013] Next, embodiments of the present invention will be described.

BRIEF DESCRIPTION OF THE DRAWINGS

[0014]FIG. 1 is a perspective view illustrating a first embodiment of apaper container according to the present invention.

[0015]FIG. 2 is a bottom face view of the paper container of FIG. 1.

[0016]FIG. 3 is a development plan of the paper container of FIG. 1.

[0017]FIG. 4 is a plan view showing a condition in which a blank formolding the paper container of FIG. 1 is extracted from raw-materialpaper.

[0018]FIG. 5 is a view showing a condition in which the blank of FIG. 1is folded up, and is a rear view as seen from FIG. 3.

[0019]FIG. 6 is an overall view of a paper container according to acalculation example.

[0020]FIG. 7 is a front view of a circular cone used in the calculation.

[0021]FIG. 8 is a development plan of the circular cone of FIG. 7.

[0022]FIG. 9 is a view illustrating a second embodiment.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0023] [First Embodiment]

[0024] Examples of application of the present invention are described indetail below with reference to the drawings. First of all, an example ofthe present invention is described with reference to FIG. 1 to FIG. 5.

[0025] [Construction of the Paper Container]

[0026]FIG. 1 is a perspective view showing an overall view of aPractical Example of a paper container. FIG. 2 shows a bottom face viewof the same. This paper container is integrally formed in a taperedtubular shape widening to some degree in the upward direction by foldingup a single-sheet blank. The paper container is constituted of a bottomface 1 and a peripheral wall 2, its upper face 3 being open. Oppositelyto the paper container as shown in FIG. 1, it could also be constitutedin an inverse tapered shape widening from upper face 3 to bottom face 1.

[0027] Although in the present embodiment bottom face 1 is constitutedby a regular dodecagonal shape, in general it could be of circular shapeor regular polygonal shape other than dodecagonal. Peripheral wall face2 is constituted of a plurality of partitioned outside divided faces(portion constituting the outer wall face) 5 and inner pleated faces onthe inner peripheral side (portion constituting the inside wall face) 4.As shown in FIG. 1, the divided faces 5 refer to the outer peripheralwall constituted of quadrilaterals, and inner pleated faces 4 refer tothe portions where the sheet is folded and overlapped in two layers.Each divided face 5 stands erect with a prescribed gradient from thebottom face 1 towards the circumferential direction, and extends alongthe outer peripheral face of peripheral wall 2 as far as the upper face3.

[0028] That is, the divided faces 5 are formed in strip shape (axiallyelongate shape) extending from the peripheral edge of bottom face 1 inhelical fashion towards the edge 3A of the opening of the upper face;their side edges 5A are mutually brought up against each other so thatthe inner pleated faces 4 there between (portions where two sheets areoverlapped) are arranged in linked fashion in the peripheral directionsandwiched between two divided faces 5. The inner pleated faces 4 areconstituted by folding over two triangular shapes (see the developmentplan of FIG. 3).

[0029] Furthermore, as is clear from FIG. 3, the inner pleated faces 4are folded over on their inside faces along respective divided faces 5,being mutually interposed between divided faces 5 in triangular fashion,folded in two, with the vertices of the triangles touching a peripheraledge 1A of bottom face 1 (corner of the dodecagon). In a paper containerconstructed from a single sheet of paper in this way, the papercontainer can be maintained in fixed shape without employing anyadhesive at all. A paper container constructed in this way can beemployed as a blank for containers for food products or plant pots etcby using coating paper formed with a covering film of synthetic resinfilm or other water-repellent material.

[0030] [Development Plan]

[0031]FIG. 3 shows this paper container in opened-out condition. In FIG.3, the hill fold lines (rule lines 7) are indicated by broken lines andthe valley fold lines (lines 9) are indicated by thin lines. Also, fromthis Figure, bottom face 1 is defined by annular rule line 6 forming aregular dodecagonal shape in the middle of blank B and branching rulelines, lines 9 and rule line 7 are provided extending in radial fashionfrom each corner of this annular rule line 6.

[0032] If the combination of a single divided face 5 and a single innerpleated face 4 is considered as the structural unit of a singleperipheral wall face 2, the number of such structural units is equal tothe number of corners of the bottom face. In the Figure, divided face 5is the quadrilateral E′ACB, and inner pleated face 4 is thequadrilateral ADHC consisting of ΔADC and ΔDHC. The lead angle of theside faces 5A of divided faces 5 of the paper container is α.

[0033] ΔADC and ΔDHC are hill-folded at rule line 7 and arevalley-folded at line 9, and overlaid on ΔHEC of the adjacent dividedface 5. The torsional angle of line AB and line DC is φ. The lead angleα of side face 5A of divided face 5 is different from the torsionalangle φ. If line DC is a straight line, the torsional angle φ will be 0.

[0034] When the paper container is produced, the quadrilateral E′ACBappears as a divided face from outside the paper container and pleatedface 4 (quadrilateral ADHC) is not visible. From within the papercontainer, quadrilateral ADCB is visible and A DHC and ΔHEC are notvisible.

[0035] Also, as can be seen from the Figure, polygon BADHEC can beconsidered as the structural unit of the wall face of the papercontainer.

[0036] Inner pleated face 4 is constituted by the overlapping portionproduced by a hill-folded line at rule line 7 centered thereon andproduced by folding defining two triangles 8 with angle φ. As is clearfrom FIG. 3, blank B constituting the paper container is defined by aregular dodecagon defining the bottom face 1 (annular rule line 6), anda larger-diameter regular dodecagon arranged concentrically therewith,its corners being linked with the corners of the regular dodecagon ofthe bottom face 1 by rule lines 7.

[0037] In the development plan, when the paper container is constructedby folding up along the hill fold lines and valley fold lines, the lineswhere rule line 7 and line 9 are superimposed are respectively indicatedin the drawing as line 7′ and line 9′.

[0038] Branching lines 9′, 9 are straight lines drawn towards the largerdiameter regular dodecagon with angle φ from the corners of the bottomface 1 on both sides of and centered on rule line 7. Inner pleated face4 is the region on the inside of branching lines 9′, 9. Consequently,the divided faces 5 are elongate quadrilaterals with one sideconstituted by each face of annular rule line 6 and extending with acertain angle to the radial direction, the inner pleated faces 4 (seeFIG. 1) being formed mutually there between. Line 7′ extends from acorner of the bottom face 1 while making an angle φ with line 9.

[0039] In FIG. 3, the angle made by the branching rule lines 7, 7′ isshown as 2φ; by varying this angle φ, the degree of opening of the upperface of the paper container obtained can be made larger or smaller.Clearly, also, as the angle θ′ of the line 7 and the radial direction ofthe bottom face 1 approaches π, the tapering of the paper containerbecomes less.

[0040] [Method of Manufacture]

[0041] A method of manufacturing a paper container constructed in thisway will now be described with reference to FIG. 4 to FIG. 5. First ofall, prescribed raw-material paper P is prepared as shown in FIG. 4, andthis is converted into a blank B by cutting to a prescribed shape, inparticular in this embodiment a regular dodecagonal shape, for exampleusing a trimming die. In particular, by using a trimming dieincorporating rule lines in addition to the cutting edges, blank B maybe formed with an annular rule line 6 and lines 9 for constitutingvalley fold lines, as well as lines 7 for constituting hill fold lines,simultaneously with the molding thereof. Lines 7 are formed so as toextend making an angle θ′ with the radial direction of the bottom face 1and lines 9 are formed on one side thereof making an angle φ with rulelines 7. Annular rule line 6 is formed in the middle of blank B in theshape of a regular dodecagon.

[0042] In this way, inner pleated faces 4 are constituted as the regionsof triangles 8 on both sides of rule lines 7 used as hill-folded lines;and inner pleated faces 4 and the strip shaped (rectangular) dividedfaces 5 formed mutually there between are alternately defined at theperiphery of annular rule line 6. FIG. 5 is a plan view showing anintermediate condition in the production of a paper container by foldingthe blank of the development plan of FIG. 3, and is a rear view as seenfrom FIG. 3. As shown in FIG. 5, the side edges 5A (back faces ofbranching lines 9, 9′) that define divided faces 5 are brought upagainst each other by folding the blank upwards along the upward-foldingbroken line 7 and folding downwards along the downward-folding thin line9 so as to fold in two each of the inner pleated faces 4. The innerpleated faces 4, which are thus folded in two are thereby overlaid onthe back faces i.e. the inner peripheral faces along the innerperipheral face side, of each of the divided faces 5.

[0043] A paper container as shown in FIG. 1 can thereby be obtained.

[0044] Also, a paper container of this type can be automatically molded(not shown) by coaxially arranging a cavity having ribs for effectingfolding-in at rule lines 7 and a punch having grooves for receiving theinner pleated faces 4 which are folded in two. In particular, as a meansfor overlaying inner pleated faces 4 on divided faces 5, considerationmay be given to indexed rotation of the punch following the helicalshape of divided faces 5, with the cavity fixed.

[0045] The rim 3A of the aperture of the upper face of the papercontainer is made level (see FIG. 1) by making the peripheral edge ofblank B flower petal shaped. The rim 3A of the upper face of the papercontainer shown in FIG. 1 may be left without any kind of processing or,as in this embodiment, the rim 3A of the upper face may be subjected tocurling in which its outside is folded back to the inside. Thepossibility of the user of the paper container being injured by contactwith the rim 3A of the aperture of the upper face is thereby reduced.

[0046] Also, the paper container can be prevented from being opened outeven in the case where the taper angle is shallow (paper container ofsmall height), by folding back, outwards or inwards by curling, the rim3A of the aperture of the upper face of the paper container obtained. Itshould be noted that, although it is possible to maintain the papercontainer in fixed shape without using any adhesive at all sincespreading out of the rim 3A of the aperture of the upper face isprevented by the fact that when the rim 3A of the aperture of the upperface is folded back outwards or inwards by for example curling acondition is maintained in which the inner pleated faces 4 are folded upalong the divided faces 5, it would also be possible to stick the innerpleated faces 4 on to the divided faces 5 by using adhesive instead offolding in the rim 3A of the aperture of the upper face.

[0047] Also, the paper containers according to the present invention arenot restricted to paper containers whose bottom face 1 is of regulardodecagonal shape as described above and bottom face 1, peripheral wallface 2 and upper face 3 could be made of substantially circular shape byfurther reducing the width of inner pleated faces 4 and divided faces 5,or these could be made of polygonal shape, such as triangular shape orquadrilateral or even twenty four gon shape, in particular, the regularpolygons of these.

[0048] [Method of Calculation]

[0049] A method of determining and calculating the various necessaryparameters for forming a paper container by the above steps will now bedescribed. In general, in almost all cases, the height of the papercontainer and the radius of bottom face 1 and upper face 3 are given; inaddition, the number of corners of bottom face 1 is often given. In somecases, as shown in FIG. 6, the lead angle a of the lateral side 5A ofthe divided face 5 or the torsional angle φ of the lateral sides AB. DCof quadrilateral ADCB seen from inside the paper container may be given.

[0050] Herein below, a method of determining torsional angle φ (θ′ or θ)and the length of the sides and angles of inner pleated faces 4 anddivided faces 5 when the height h₁ of the paper container, radius r₁ ofbottom face 1, radius r₂ of upper face 3 and bottom face 1 is given as aregular n-gon are given as initial conditions is described. A papercontainer molded in accordance with the parameters determined by acalculation as below was found to be fully satisfactory for manufactureas a paper container within the range of manufacturing error.

[0051] A method of calculating the various structural elements of thepaper container will be described with reference to FIG. 3 and FIG. 6 toFIG. 8. In general, an development plan can be obtained if the radius r₂of the upper face 3 of the paper container, the radius r₁ of its bottomface 1, the height h₁ of the paper container, the number of corners n ofbottom face 1 and the torsional angle φ (are given.

[0052]FIG. 6 is an overall view of the paper container and FIG. 3 is adevelopment plan thereof. The number of divided faces 5 of peripheralwall face 2 is the same as the number of sides of bottom face 1;overall, the paper container is formed so as to be tapered in helicalfashion with an angle φ. These divided faces 5 have an axially elongatequadrilateral shape (polygon E′ACB) from each side of regulardodecagonal bottom face 1 outwards in the radial direction in thedevelopment plan, FIG. 3. In the development plan, the region betweenone divided face 5 and another divided face 5 constitutes an innerpleated face 4 that is folded in two (quadrilateral ADHC (consisting ofΔADC and ΔDHC); inner pleated face 4 is folded over along the hill-foldlines and valley-fold lines so as to overlie divided face 5.

[0053] To achieve this, it is necessary for ∠ HCD and ∠ ECH to be equalangles φ. Also, when the paper container is produced, in order for thedivided faces 5 and inner pleated faces 4 to overlap uniformly (in atriply overlapping condition seen from any part of the upper face of theaperture of the paper container), it is necessary that line sectionsE′A, AD, DH, and HE should respectively be equal. To achieve this, it isnecessary that ∠DCA=/HCD=∠ECH=φ. Also, if the length of each side of thequadrilateral ΔDCB and ΔDHC and ΔHEC, and each angle are found, adevelopment plan of the paper container can be obtained, enabling thepaper container to be produced.

[0054] The method of determination and calculation of the variousparameters of the quadrilateral ΔDCB and ΔDHC and ΔHEC that arenecessary when manufacturing the paper container will now be describedin detail. Since, if the bottom face one of the paper container is ofpolygonal shape and the number of corners n is sufficiently large, itcan be approximated as a conical shape, it will be examined in terms ofthis form.

[0055] Cutting is effected at a plane including the centerline of thepaper container of centerline height h₁ that is to be manufactured. Theline extending the generating line 103 represented at thecross-sectional plane when the cut is made and the line extending thecenterline of the paper container intersect at T. That is, if the bottompart of the paper container is extended, it becomes a circular conicalshape, the aforementioned cross-sectional plane being the shape obtainedby cutting this. The vertex 102 when the peripheral wall 2 of theconical shape is extended in the direction of the bottom face 1 as shownin FIG. 7 to define a right circular cone 101 will be designated as T.The paper container according to the present invention may be describedas a shape equal to that obtained by cutting this right circular cone101 in a direction at right angles to a given axis.

[0056]FIG. 8 is a development plan of this circular cone 101. In FIG. 7,the height of the cone 101 defined by the upper face 3 of the containerand vertex T is H, the length of generating line 103 is l₂, the heightof cone 104 defined by bottom face 1 and vertex T is h₂, and that ofgenerating line 105 is l₁.

[0057] Let ∠DAB of polygon ABCD 106 be ∠A, ∠B, ∠C, and ∠D being definedin like fashion. In order to create a development plan of the containerfrom the initial parameters n, r₁, r₂, and h₁ (in the case of a uniformupper face 3), the lengths of each side and the angles and value φ ofquadrilateral ADCB 106 are required.

[0058] [Calculation of φ]

[0059] [Math 7]

∠B+∠C+2∠OBC+2φ=2π

[0060] From the law of the internal angles of a quadrilateral and fromΔABT and ΔDCT of FIG. 8,

[0061] [Math 8]

φ=∠TAD−∠OBC=(½−r ₂ /nl ₂)π−(½−1/n)π=(1−r ₂ /l ₂)π/n

[0062] where

[0063] [Math 9]

l ₂ =TA={square root}{square root over ((H ² +r ₂ ²))}

H=h ₁ +h ₂ =h ₁ +r ₁ h ₁/(r ₂ −r ₁)

[0064] φ is therefore uniquely determined by n, r₁, r₂ and h₁.

[0065] [Calculation of Sides]

[0066] The lengths of the sides of quadrilateral ADCB 106 are calculatedfrom the expressions given below.

[0067] [Math 10]

arcAD=2πr ₂ /n  (1)

AD=2l ₂ sin(πr ₂ /nl ₂)  (2)

arcBC=2πr ₁ /n  (3)

BC=2r ₁ sin(π/n)  (4)

[0068] Length of hill-fold line 7 (side DC):

AB=CD={square root}{square root over ((l ₁ ² +l ₂ ²−2l ₁ l ₂ cosθ))}  (5)

[0069] where θ=∠BTA=φr₂/l₂

[0070] [Calculation of Angles]

[0071] Also, the angles of the quadrilateral ADCB 106 are calculated asfollows:

[0072] [Math 11] $\begin{matrix}{{\angle \quad A} = {{{\angle \quad {TAD}} + {\angle \quad {TAB}}} = {{\left( {\frac{1}{2} - \frac{r2}{nl2}} \right)\pi} + {\arccos \left( \frac{L^{2} + l_{2}^{2} - l_{1}^{2}}{2{Ll}_{2}} \right)}}}} & (1) \\{{\angle \quad B} = {{{\angle \quad {TBA}} - {\angle \quad {TBC}}} = {{\arccos \left( \frac{L^{2} + l_{1}^{2} - l_{2}^{2}}{2\quad {Ll}_{1}} \right)} - {\arccos \left( {\frac{r_{1}}{l_{1}}\sin \quad \frac{\pi}{n}} \right)}}}} & (2) \\{{{\angle \quad D} = {{{\angle \quad {TAD}} - {\angle \quad {TAB}}} = {{\left( {\frac{1}{2} - \frac{r2}{nl2}} \right)\pi} - {\arccos \left( \frac{L^{2} + l_{2}^{2} - l_{1}^{2}}{2{Ll}_{2}} \right)}}}}{{\angle \quad C} = {{2\quad \pi} - {\angle \quad A} - {\angle \quad B} - {\angle \quad D}}}} & (3)\end{matrix}$

[0073] where

L=AB=CD={square root}{square root over ((l ₁ ² +l ₂ ²−2l ₁ l ₂ cos θ))}

θ=∠BTA=r₂ φ/l ₂

∠ATD=2πr ₂ /nl ₂

∠TAD=(½−r ₂ /nl ₂)π

∠TBC=arccos((r₁ /l ₁)sin(π/n))

∠OBC=(½−1/n)π

[0074] The angle between the radius r₁ of the regular polygon of bottomface 1 and AB is

[0075] [Math 12]

∠OBA=φ′=∠OBC+∠B

[0076] As is clear from the above calculation, the development plan canbe obtained if n, r₁, r₂, h₁ and θ or φ are given. Furthermore, φ isindependent from φ, and if the values of r₁, r₂ and h₁ are given, apaper container of the same shape can be produced using a differentvalue of φ. The condition of the paper at the rim of the uppermost faceof the paper container can be made to be a single sheet, or threesheets, or, if appropriate, five sheets, at particular locations.

[0077] [Calculation When the Edge Sides on the Upper Face Side of theDivided Faces Triply Overlap]

[0078] Also, when the condition that n, r₁, r₂, h₁ and the rim 3A of theaperture of the upper face triply overlap is inserted as an initialcondition for the paper container, it is found to be necessary that

[0079] ∠ACD=φ and

[0080] AC=HC

[0081] The method of calculation in this case is indicated below.

[0082] When equations are written for A, B, C and D, the followingdeterminant is obtained. Putting

P₁=A

P₂=C

P₃=T

P₄=D

d _(ij) =P _(i) P _(j)

[0083] and putting AC=d₁₂=x, d₁₃=l₂,

[0084] [Math 13]

[0085] we have

d ₁₄=2l ₂ sin(πr ₂ /nl ₂)

d₂₃=l₁, d₂₄=L,

d₃₄=l₂.

[0086] [Math 14]

d ₂₄ =L={square root}{square root over ((l ₁ ² +l ₂ ²−2l ₁ ^(l) ₂ cosθ))}

[0087] is a variable of θ.

[0088] Apart from d₁₂ and d₂₄, this is uniquely determined by n, r₁, r₂and h₁.

[0089] The following determinant is obtained. $\begin{matrix}{M = \begin{pmatrix}0 & d_{12}^{2} & d_{13}^{2} & d_{14}^{2} & 1 \\D_{12}^{2} & 0 & d_{23}^{2} & d_{24}^{2} & 1 \\d_{13}^{2} & d_{23}^{2} & 0 & d_{34}^{2} & 1 \\D_{14}^{2} & d_{24}^{2} & d_{34}^{2} & 0 & 1 \\1 & 1 & 1 & 1 & 0\end{pmatrix}} & \left\lbrack {{Math}\quad 15} \right\rbrack\end{matrix}$

[0090] Since point A, point C, point T and point D are on the sameplane, the determinant M is 0.

[0091] Therefore

det(M)=0  (equation C)

[0092] The relationship expression for ∠ACD=φ is as follows:

[0093] [Math 16]

(L ² +x ² −AD ²)/2Lx=cos φ

[L ² +x ²−{2l ₂ sin(πr ₂ /nl ₂)}²]/2Lx=cos [[1−r ₂ /l₂](π/n)]  (equation D)

[0094] which is an equation in the two variables x and θ.

[0095] θ can be obtained by solving the simultaneous equations: equationC and equation D.

[0096] From the value of θ, [Math 17]

θ=∠BTA=φr₂ /l _(2.)

[0097] the value of φ scan also be found by the equation:

[0098] Also, the value φ can be found by directly, without going throughθ, by rewriting the equation.

[0099] In this way, the length of AC can be calculated.

[0100] [Example of Method of Constructing a Development Plan]

[0101] First of all, a regular n-gon of radius r₁ defining the bottomface 1 is constructed, and n triangles are constructed linking eachvertex thereof and the center point O of the polygonal shape of thebottom face 1. A quadrilateral ADCB is then constructed from theseinterior triangles in the radially outwards direction. In doing this,the angles and sides of ABCD obtained by calculation are utilized. Theline CH making an angle φ therewith is constructed, and the polygonBADHC is thereby obtained.

[0102] The next polygon can be constructed by shifting this polygonBADHC through an angle 2 π/n about the center point O. By repeating thisstep, a development plan of the paper container is obtained and thepaper container can be constructed by hill-folding and valley-foldingalong the respective lines. In order to obtain polygon BADHC, the lengthof AB, the length of AD, and the values of φ and φ′ are necessary; thesevalues are calculated by the above formulae from the initial conditionsn, r₁, r₂, h₁.

[0103] Formation of the development plan is not restricted to using thesequential steps described above but could be achieved by any sequenceusing the calculated lengths of the various sides and of the variousangles.

[0104] [Second Embodiment]

[0105]FIG. 9(a) and (b) show a second embodiment of a paper containerwherein curling is performed at the upper face, FIG. 9(a) being a planview thereof and FIG. 9(b) being a plan view of FIG. 9(a) with partbroken away. Opening out of the divided faces 22 of paper container 20is prevented by curling 21 of the upper edge of paper container 20. Ifthe lead angle α of the lateral sides 5A of the divided faces 5described above or the torsional angle φ of the lateral sides AB and CDof the quadrilateral ADCB seen from within the paper container arecomparatively large, a paper container can be constructed wherein thedivided faces 22 are not easily opened out.

[0106] As can be seen from FIG. 9(b), only part of the rim of dividedfaces 22 on the upper face side of paper container 20 constitutes ablank which is triply overlaid. In this embodiment, curling 21 isperformed in order to prevent opening out of the rim of the dividedfaces on the upper face side. However, it is possible to construct apaper container 20 in which the divided faces 22 are opened out withoutapplying curling to the rim of divided faces 22 of paper container 20 onthe upper face side.

[0107] As is clear from the above description, with the presentinvention, a single blank can be formed in tubular shape, leaving itsmiddle part intact, by forming pleats by gusset folding of the peripherythereof, so a paper container with a deep bottom can easily beconstructed without damaging the blank; thus a distinction can beachieved over conventional plastic containers.

[0108] Also, since this paper container can be formed with a deepbottom, its possible applications are expanded; in particular, since itis integrally molded from a single blank, by employing coated paper forthe blank, in contrast to paper containers obtained by the paper-makingmethod, it can be given waterproof properties such as make possible itsapplication even to drinks containers. Furthermore, since it has innerpleated faces that are folded up in the peripheral face, it has highstrength and good appearance. Moreover, the fixed shape can bemaintained without use of adhesive, by subjecting the rim of the upperface aperture to curling.

What is claimed is:
 1. A paper container which is integrally formed froma single-sheet blank and the upper face of which is open, said papercontainer comprising: a polygonal bottom face (1); and a peripheral wallface (2) consisting of a plurality of outside divided faces (5) ofhelically wound shape and of inner pleated faces (4) constituting aninner wall face by being folded in two on the inside and continuouslyoverlaid; wherein, in the development plan of said paper container, thebottom face (1) is positioned at the center of the single-sheet blank;said divided faces (5) of quadrilateral shape and said inner pleatedfaces (4) consisting of two triangles (8) are provided at the peripheryof said bottom face (1) in a number equal to the number of sides of saidbottom face (1); said divided faces (5) and said inner pleated faces (4)are positioned alternately and extend in linear fashion from theperipheral edge of said bottom face (1) towards the outside in theradial direction; the blank portion between one said divided face (5)and another said divided face (5) constitutes said inner pleated face(4), whose vertex is a corner vertex of said bottom face (1); said innerpleated face (4) consists of two triangles (8) having as common vertex acorner of said bottom face (1) and a common side which is axis ofsymmetry (7); and said inner pleated faces (4) are overlapped on theinside of said divided face (5) by folding up on said axes of symmetry(7).
 2. A method of manufacturing a paper container which is integrallyformed from a single-sheet blank, its upper face (3) being open, saidpaper container comprising a polygonal bottom face (1), and a peripheralwall face (2) consisting of a plurality of outside divided faces (5) ofhelically wound shape and of inner pleated faces (4) constituting aninner wall face by being folded in two on the inside and continuouslyoverlaid, wherein, in the development plan of said paper container, saidbottom face (1) is positioned at the center of the single-sheet blank;said divided faces (5) of quadrilateral shape and said inner pleatedfaces (4) consisting of two triangles (8) are provided at the peripheryof said bottom face (1) in a number equal to the number of sides of saidbottom face (1); said divided faces (5) and said inner pleated faces (4)are positioned alternately and extend in linear fashion from theperipheral edge of said bottom face (1) towards the outside in theradial direction; the blank portion between one said divided face (5)and another said divided face (5) constitutes said inner pleated face(4), whose vertex is a corner vertex of said bottom face (1); said innerpleated face (4) consists of two triangles (8) having as common vertex acorner of said bottom face (1) and a common side which is axis ofsymmetry (7); and a paper container is manufactured by folding up saidinner pleated faces (4) on said axes of symmetry (7) and overlappingsame on the inside of said divided face (5). and thereby manufactured.3. A method of manufacturing a paper container which is integrallyformed from a single-sheet blank, its upper face (3) being open, saidpaper container comprising a polygonal bottom face (1), and a peripheralwall face (2) consisting of a plurality of outside divided faces (5) ofhelically wound shape and of inner pleated faces (4) constituting aninner wall face by being folded in two on the inside and continuouslyoverlaid, wherein, in the development plan of said paper container, saidbottom face (1) is positioned at the center of the single-sheet blank;said divided faces (5) of quadrilateral shape and said inner pleatedfaces (4) consisting of two triangles (8) are provided at the peripheryof said bottom face (1) in a number equal to the number of sides of saidbottom face (1); said divided faces (5) and said inner pleated faces (4)are positioned alternately and extend in linear fashion from theperipheral edge of said bottom face (1) towards the outside in theradial direction; the blank portion between one said divided face (5)and another said divided face (5) constitutes said inner pleated face(4), whose vertex is a corner vertex of said bottom face (1); said innerpleated face (4) consists of two triangles (8) having as common vertex acorner of said bottom face (1) and a common side which is axis ofsymmetry (7); and the angle φ of the common vertex of said two triangles(8) and the sides of said divided face (5) are respectively calculatedby the following formulae: Calculation formulae: [Math 1] φ=[1−r ₂ /l₂](π/n) l ₂={square root}{square root over ((H ² +r ₂ ²))}H=h ₁ +h ₂ =h₁ +r ₁ h ₁/(r ₂ −r ₁) l ₁={square root}{square root over ((h ₂ ² +r ₁²))}|length of side on upper face side (3A) of divided face (5)|=2l ₂sin(πr ₂ /nl ₂) |length of side on bottom face side (1) of divided face(5)|=2r ₁ sin(π/n) |length of lateral side of divided face (5)|={squareroot}{square root over ((l ₁ ² +l ₂ ²−2l ₁ l ₂ cos θ))} where θ=φr₂/l₂,h₂=r₁/r₂−r₁ when h₁ is the height of the paper container, r₂ is theradius of upper face (3), r₁ is the radius of bottom face (1), n is thenumber of corners of bottom face (1).
 4. The method of manufacturing apaper container according to claim 2 or claim 3, wherein the edge side(3A) on the side of said upper face (3) of said divided face (5) iscalculated by the following formulae in order to achieve triple overlap.[Calculation When There is Triple Overlaps of the Edge Sides on theUpper Face Side] (where h₁ is the height of the paper container, r₂ isthe radius of upper face (3), r₁ is the radius of the bottom face (1), nis the number of corners of bottom face (1), quadrilateral E′ACB isdivided face (5), E′B and AC are the lateral sides of divided face (5),E′A is the edge side on the side of upper face (3) of divided face (5),BC is the edge side on the side of bottom face (1) of the divided face(5), polygon ADHECB is the structural unit of the peripheral faceconstituting the paper container (the development plan of the papercontainer is constructed from bottom face (1) and n polygons ADHECBaround this), φ is the torsional angle of line AB and line DC, ∠ACD=φ ishalf of the angle 2φ of the inner pleated face (4) extending from acorner of the bottom face, and T is the vertex (T) when the bottom face(1) side of the paper container is extended to be developed as cone(101) Condition for triple overlap: assuming ∠ACD=φ, AC=HC and that thevertices of the divided side (5) and T are: P₁=A P₂=C P₃=T P₄=D then d_(ij) =P _(i) P _(j) AC=d₁₂=x d₁₃=l₂, [Math 2] d ₁₄=2l ₂ sin(πr ₂ /nl ₂)d₂₃=l₁, d₂₄=L, d₃₄=l₂ [Math 3] where L={square root}{square root over((l ₁ ² +l ₂ ²−2l ₁ l ₂ cos θ))} and apart from d₁₂ and d₂₄, this isuniquely determined by n, r₁, r₂ and h₁. Writing the equations, thefollowing matrix is obtained: $\begin{matrix}{M = \begin{pmatrix}0 & d_{12}^{2} & d_{13}^{2} & d_{14}^{2} & 1 \\D_{12}^{2} & 0 & d_{23}^{2} & d_{24}^{2} & 1 \\d_{13}^{2} & d_{23}^{2} & 0 & d_{34}^{2} & 1 \\D_{14}^{2} & d_{24}^{2} & d_{34}^{2} & 0 & 1 \\1 & 1 & 1 & 1 & 0\end{pmatrix}} & {\left\lbrack {{Math}\quad 4} \right\rbrack \quad\left\lbrack {{Math}\quad 15} \right\rbrack}\end{matrix}$

Since point A, point C, point T and point D are on the same plane, thedeterminant M is
 0. Therefore det(M)=0  (equation A) The relationshipexpression for ∠ACD=φ is as follows: [Math 5] (L ² +x ² −AD ²)/2Lx=cosθ[L ² +x ²−{2l ₂ sin(πr ₂ /nl ₂)}²]/2Lx=cos [[1−r ₂ /l₂](π/n)]  (equation B) which is an equation in the two variables x andθ. θ can be obtained by solving the simultaneous equations: equation Aand equation B. From the value of θ, [Math 6] θ=∠BTA=r₂ /l ₂ the valueof φ can also be found by the equation: and the value of φ can beobtained by directly writing the equation without going via θ.Accordingly, the length of AC can be calculated and the development planof the paper container uniquely found.
 5. The method of manufacturing apaper container according to claim 2 or claim 3, wherein the aperturerim (3A) of said upper face is produced by curling.